Over the past 40 years, he has amassed a collection of more than 20,000 Islamic art objects, dating from the eighth to the 20th centuries, ranging from miniature paintings and illustrated manuscripts to textiles, ceramics and metalwork. He’s still buying: In the last two years, the Khalili Family Trust has spent about £30 million ($47.4 million) on Islamic art.

But it’s not so easy any more. “Twenty years ago, there would be 20 magnificent pieces and four people buying,” he says. “Now there will be four major, important pieces if you’re lucky and 50 people buying.”

The market for Islamic art has been boiling. Even in the United States, where the field has been neglected since the 1979 Iranian revolution, it’s gaining buyers. Last November, the Metropolitan Museum of Art opened new galleries for Islamic art that in less than eight months attracted 593,000 visitors and helped boost annual attendance to a record-setting 6.3 million. “The Met created huge interest,” says Simon Ray, a London dealer.

Around the globe, other museums are also spotlighting Islamic art and augmenting their collections. On Sept. 22, the Louvre will inaugurate a grand pavilion for Islamic art, and next year the Aga Khan Museum will debut in Toronto.

The Gulf States have started or are planning museums, public and private—most notably the five-year-old Museum of Islamic Art in Doha. “They drove the market in the last 10 years,” says Brendan Lynch, of London dealers Oliver Forge and Brendan Lynch Ltd.

But he and others believe there is much more buying ahead. “I feel strongly that Islamic art is undervalued in almost every area,” says Edward Gibbs, head of the Middle East Department at Sotheby’s. For example, “you could build a sensational collection of medieval Islamic ceramics—a world-class, museum-quality collection—for below their long-term value,” he says. That’s not true of many other sectors of the art market, where great works are scarce or the prices are enormous.

The record price at auction for an Islamic work of art is just under £7.4 million for an illustrated folio from the famed Shahnameh, a 16th-century manuscript chronicling Persian history, which sold at Sotheby’s in London in April 2011. That’s a small fraction of the prices that many contemporary and modern works of art have been fetching. (All auction prices include the commission that buyers must pay; estimates do not.)

The center of the Islamic art market is London, where the biggest dealers are located and where Christie’s, Sotheby’s and Bonhams (a smaller auction house) hold auctions of Islamic art each spring and fall.

Though sales are growing, totals have varied considerably from year to year. Spring 2011 marked a peak in volume: Combined sales at Sotheby’s and Christie’s jumped to $78.9 million, plumped by interest in the collection of Stuart Cary Welch, a renowned American scholar of Islamic art, and in pieces from the collection of Simon Digby, an English orientalist.

Last spring, the combined total fell to $23.3 million, partly because there were fewer objects and no star lots. Some buyers felt that some estimates were too high and that some items had been on the market too recently. At Sotheby’s, just over half the lots sold. But Mr. Gibbs said that “after the sale, we sold more than 90% of the unsold lots, most at or above” the floor price. At Christie’s much smaller sale, about 70% of the lots sold.

Some critics blame the auction houses for pushing the market too aggressively. But William Robinson, Christie’s international head of Islamic Art, cites another factor: “It’s quite an emotional market. When things are going well, prices go way over the top estimate. Six months later, an identical object gets no response. Or vice versa.”

As an example, he cites a 13th-century Seljuk ewer he offered in October 2010, with an estimate of £50,000 to £80,000. It didn’t sell. Recycled into the April 2011 sale, with an estimate of £30,000 to £50,000, it soared to £361,250. “It shows how difficult pricing is in this market,” he says.

That may be changing now, as the market broadens and loses its reputation as the province of secretive Gulf sheiks and European aesthetes.

A version of this article appeared September 1, 2012, on page C14 in the U.S. edition of The Wall Street Journal, with the headline: An Ever-Busier Bazaar for Islamic Art.